Topological Quality of Subsets via Persistence Matching Diagrams

TL;DR

This paper introduces a topological data analysis method called persistence matching diagrams to evaluate the quality of data subsets, aiding in understanding their representativeness and potential impact on machine learning performance.

Contribution

It presents a novel topological invariant and an efficient algorithm to assess subset quality and its relation to the full dataset, improving data selection strategies.

Findings

Persistence matching diagrams effectively measure subset representativeness.

The method provides bounds for Hausdorff distance between subset and dataset.

Application explains subset quality's impact on model performance.

Abstract

Data quality is crucial for the successful training, generalization and performance of machine learning models. We propose to measure the quality of a subset concerning the dataset it represents, using topological data analysis techniques. Specifically, we define the persistence matching diagram, a topological invariant derived from combining embeddings with persistent homology. We provide an algorithm to compute it using minimum spanning trees. Also, the invariant allows us to understand whether the subset ``represents well" the clusters from the larger dataset or not, and we also use it to estimate bounds for the Hausdorff distance between the subset and the complete dataset. In particular, this approach enables us to explain why the chosen subset is likely to result in poor performance of a supervised learning model.